Subgroups — #16845308

Order 3968.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521454	C₂	1	1	yes	maximal, characteristic
2	#14521447	C₂	1	2	yes	maximal
3	#14521429	C₂	1	1	yes	maximal, characteristic
4	#14521400	C₂	1	2	yes	maximal
5	C₈²×C₃₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₂₄	—	2	2	no	—
7	#11758446	C₂²	1	2	yes	—
8	#11758446	C₂²	1	2	yes	—
9	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	#11758406	—	2	2	no	—
13	#11758405	—	2	4	no	—
14	#11758404	—	2	4	no	—
15	C₂×Dic₆₂	—	2	2	no	—
16	C₂×Dic₆₂	—	2	2	no	—
17	C₂×C₈×C₃₁	D₄	1	2	yes	—
18	C₂×C₈×C₃₁	D₄	1	2	yes	—
19	C₂×C₈×C₃₁	D₄	1	2	yes	—
20	C₄²×C₃₁	C₂³	1	1	yes	characteristic
21	C₄⋊Dic₃₁	—	2	2	no	—
22	C₄⋊Dic₃₁	—	2	2	no	—
23	C₄⋊Dic₃₁	—	2	2	no	—
24	C₄⋊Dic₃₁	—	2	2	no	—
25	Dic₁₂₄	—	4	4	no	—
26	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
27	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
28	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
29	C₂×Dic₃₁	—	4	2	no	—
30	C₂×Dic₃₁	—	4	2	no	—
31	Dic₆₂	—	4	4	no	—
32	Dic₆₂	—	4	4	no	—
33	C₈×C₃₁	—	2	2	no	—
34	C₈×C₃₁	SD₁₆	1	4	yes	—
35	C₈×C₃₁	Q₁₆	1	4	yes	complemented
36	#1707	—	31	1	no	maximal, Sylow
37	C₂²×C₃₁	#127	1	1	yes	characteristic
38	C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
39	C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
40	C₄×C₃₁	#135	1	2	yes	—
41	Dic₃₁	—	8	2	no	—
42	Dic₃₁	—	8	2	no	—
43	#500	—	31	1	no	—
44	#498	—	31	2	no	—
45	#494	—	31	1	no	—
46	#487	—	31	2	no	—
47	C₈²	D₃₁	1	1	yes	characteristic
48	C₂×C₃₁	#492	1	1	yes	characteristic
49	C₂×C₃₁	#494	1	1	yes	characteristic
50	C₂×C₃₁	#495	1	1	yes	characteristic
51	C₂×Q₁₆	—	62	2	no	—
52	C₄⋊Q₈	—	31	2	no	—
53	C₄⋊Q₈	—	31	2	no	—
54	#107	—	62	2	no	—
55	#106	—	62	4	no	—
56	#103	—	62	4	no	—
57	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
58	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
59	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
60	C₃₁	#1707	1	1	yes	characteristic, complemented, Sylow
61	C₂×Q₈	—	62	2	no	—
62	C₂×Q₈	—	62	2	no	—
63	Q₁₆	—	124	4	no	—
64	C₂×C₈	D₁₂₄	1	2	yes	—
65	C₂×C₈	D₁₂₄	1	2	yes	—
66	C₂×C₈	D₁₂₄	1	2	yes	—
67	C₄⋊C₄	—	62	2	no	—
68	C₄⋊C₄	—	62	2	no	—
69	C₄⋊C₄	—	62	2	no	—
70	C₄⋊C₄	—	62	2	no	—
71	C₄²	C₂²×D₃₁	1	1	yes	characteristic
72	Q₈	—	124	4	no	—
73	Q₈	—	124	4	no	—
74	C₂×C₄	—	124	2	no	—
75	C₂×C₄	—	124	2	no	—
76	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
77	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
78	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
79	C₈	—	2	2	no	—
80	C₈	#92470	1	4	yes	—
81	C₈	Dic₁₂₄	1	4	yes	complemented
82	C₂²	#11758451	1	1	yes	characteristic
83	C₄	—	248	2	no	—
84	C₄	—	248	2	no	—
85	C₄	C₂×#92470	1	2	yes	—
86	C₄	#11758481	1	2	yes	—
87	C₄	C₂×Dic₁₂₄	1	2	yes	—
88	C₂	#14521422	1	1	yes	characteristic
89	C₂	#14521429	1	1	yes	characteristic
90	C₂	#14521433	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11758451
commutator, cosocle	C₄²×C₃₁	C₂³
Frattini	C₄²	C₂²×D₃₁
Fitting	C₈²×C₃₁	C₂
socle, prime radical	C₂²×C₃₁	#127
hypercenter	C₈²	D₃₁
hypocenter	C₃₁	#1707
LCS	G ▷ C₄²×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
35. C₈×C₃₁	Q₁₆	semidirect
60. C₃₁	#1707	semidirect
81. C₈	Dic₁₂₄	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#14521454	2
#14521447	2
#14521429	2
#14521400	2
C₈²×C₃₁	2
#1707	31